In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point and a line. The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface. The graph of a quadratic function y=ax²+bx+c is a parabola if a≠0, and, conversely, a parabola is the graph of a quadratic function if its axis is parallel to the y-axis. The line perpendicular to the In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point and a line. The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface. The graph of a quadratic function y=ax²+bx+c is a parabola if a≠0, and, conversely, a parabola is the graph of a quadratic function if its axis is parallel to the y-axis. The line perpendicular to the In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point and a line. The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface. The graph of a quadratic function y=ax²+bx+c is a parabola if a≠0, and, conversely, a parabola is the graph of a quadratic function if its axis is parallel to the y-axis. The line perpendicular to the In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point and a line. The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic s
2. The parabola is the locus of all points in a plane that are
the same distance from a line in the plane, the directrix,
as from a fixed point in the plane, the focus.
Point Focus = Point Directrix
PF = PD
The parabola has one axis of
symmetry, which intersects
the parabola at its vertex.
| p |
The distance from the
vertex to the focus is | p |.
The distance from the
directrix to the vertex is also | p |.
The Parabola
| p |
3. For a parabola with the axis of symmetry parallel to
the y-axis and vertex at (h, k), the standard form is …
• The equation of the axis of symmetry is x = h.
• The coordinates of the focus are (h, k + p).
• The equation of the directrix
is y = k - p.
• When p is positive,
the parabola opens upward.
• When p is negative,
the parabola opens downward.
(x - h)2 = 4p(y - k)
The Standard Form of the Equation with Vertex (h, k)
4. For a parabola with an axis of symmetry parallel to the
x-axis and a vertex at (h, k), the standard form is:
• The equation of the axis of symmetry is y = k.
• The coordinates of the focus
are (h + p, k).
• The equation of the directrix
is x = h - p.
(y - k)2 = 4p(x - h)
• When p is negative, the parabola
opens to the left.
• When p is positive, the parabola
opens to the right.
The Standard Form of the Equation with Vertex (h, k)
5. Finding the Equations of Parabolas
Write the equation of the parabola with a focus at (3, 5) and
the directrix at x = 9, in standard form and general form
The distance from the focus to the directrix is 6 units,
therefore, 2p = -6, p = -3. Thus, the vertex is (6, 5).
(6, 5)
The axis of symmetry is parallel to the x-axis:
(y - k)2 = 4p(x - h) h = 6 and k = 5
Standard form
(y - 5)2 = 4(-3)(x - 6)
(y - 5)2 = -12(x - 6)
6. Find the equation of the parabola that has a minimum at
(-2, 6) and passes through the point (2, 8).
The axis of symmetry is parallel to the y-axis.
The vertex is (-2, 6), therefore, h = -2 and k = 6.
Substitute into the standard form of the equation
and solve for p:
(x - h)2 = 4p(y - k)
(2 - (-2))2 = 4p(8 - 6)
16 = 8p
2 = p
x = 2 and y = 8
(x - h)2 = 4p(y - k)
(x - (-2))2 = 4(2)(y - 6)
(x + 2)2 = 8(y - 6) Standard form
Finding the Equations of Parabolas
7. Find the coordinates of the vertex and focus,
the equation of the directrix, the axis of symmetry,
and the direction of opening of 2x2 + 4x - 2y + 6 = 0.
2x2 + 4x - 2y + 6 = 0
2(x2 + 2x + _____) = 2y - 6 + _____
1 2(1)
2(x + 1)2 = 2(y - 2)
(x + 1)2 = (y - 2)
The parabola opens to upward.
The vertex is (-1, 2).
The focus is ( -1, 2 ¼ ).
The Equation of directrix is y = 1¾ .
The axis of symmetry is x = -1 .
4p = 1
p = ¼
Analyzing a Parabola
